Kruskal, C. P. and Snir, M. (1986) A unified theory of interconnection network structure. Theoret. Comp. Sci., 48, 75--94.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in Delta MINs this control sequence is a representation of the destination. In this paper Network(N,r) will present a MIN of size N and degree r. In fact, a network having the Delta property possesses some kind of regularity so that the network s routing algorithm can be simple and well defined[13]. Definition 3 We call an over sized MIN of size N a banyan Delta MIN composed of more than one copy of a Delta MIN gathered together by an interconnection stage having the Delta property. Non banyan networks can be constructed either by the augmentation of a banyan network or by the construction ....

C.P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75-94, 1986.

....multistage routing networks. In this section, we describe these strategies and show an example of each A dilated, non interwired four stage network connecting 16 endpoints. Each component is a 4 x 2, dilation 2 crossbar. Figure 2: Dilated Non Interwired Network network. As with all bidelta [11] networks, each stage in these multipath networks routes by successively subdividing the set of possible destinations into a number of equivalence classes equal to the radix of the routing components. For example, consider a radix 4 network. When connections enter the network, any input can reach ....

....5. 3.1 Dilated Non Interwired Network The dilated networks described in [10] 6] connect all outputs in a given logical direc tion to the same physical routing component in the subsequent stage of the network. The topology is thus identical to the corresponding non dilated bidelta network [11]. Such networks gain performance by having multiple paths. These networks can also tolerate some faults in the wiring between components. Unfortunately, these networks are just as susceptible to component failures as non multipath networks. An example of a dilated non interwired network composed ....

Clyde P. Kruskal and Marc Snir. A unified theory of interconnection network structure. In Theoretical Computer Science, pages 75-94, 1986.

....and the high order n Gamma j bits will be the low order n Gamma j bits of the destination address. Thus, when the message reaches its destination, the path descriptor field may be reversed and used in the same way as the return address to the origin of the message. Following the definition in [76], in a delta network the path descriptors associated with different paths leading to the same output node are identical, so that, if the inputs are processors and the outputs are memory modules, each processor uses the same routing tag for a given memory module. Bidelta networks have this property ....

Clyde P. Kruskal and Marc Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48(1):75--94, 1986.

....which travel through interconnection networks. As the size of a system is increased to obtain greater computing power, the impact of interconnection networks on system performance and cost becomes more significant. Among various interconnection network architectures, shuffleexchange networks [2, 10, 13, 14] have been one of the most popular architectures. The networks have distributed selfrouting capabilities, and when network size is increased, This work was supported by NASA Ames Research Center under Grant No. NASA NCC 2 559, the Department of Energy under Grant No. DE FG02 85ER25001, the ....

Clyde P. Kruskal and Marc Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48(1):75--94, August 1986.

....of routing can be used to route to the desired destination. We do not concern ourselves with fault tolerance issues outside of the network. The design presented is applicable across the wide range of networks constructed using multiple routing stages, including all kinds of banyan like networks [10] and fat tree networks [13] 6] 3] 20.2 Background 20.2.1 Network Processor Interface In order for the network to be useful in the context of a large scale parallel computer, it must interface coherently with the network endpoints. For a large parallel computer, each endpoint will consist of ....

....first stage of routing. The bandwidth between network stages must match. The bandwidth out of the network must match the output bandwidth to the endpoints. The number of processors is usually a power of the radix of the crossbar routers. Square networks (i.e. n = m) are often good configurations [10], especially when all endpoints are being treated equally. Square networks are usually constructed from square crossbar routing elements (i.e. i = r Theta d) Bandwidth matching is moderately easy in these cases. Rectangular networks with n m are often desirable because they offer less network ....

Clyde P. Kruskal and Marc Snir. A unified theory of interconnection network structure. In Theoretical Computer Science, pages 75--94, 1986.

....in the context of routing and sorting. In this thesis, we limit our attention to the problem of constructing efficient sorting circuits, or sorting networks. For an introduction to routing circuits (also often referred to as interconnection networks) and a survey of results, we refer the reader to [51, 87, 108]. A 2 input comparator is a device that compares two integer values supplied on its input wires, and then outputs the larger of the two values on the output wire labeled as the maximum output, and the smaller value on the output wire labeled as the minimum output. A 2 input switch is a device ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....the hypercube itself. 1. Introduction Bounded Degree Hypercube Approximations. Various bounded degree graph families are derived from the hypercube in an attempt to achieve the efficiency of the hypercube while keeping the graph degree constant and low. The families of butterfly like graphs (cf. [21, 15]) and shuffle like graphs (cf. 25, 24, 10] have a low diameter and high bisection width, which enable them to execute algorithms which demand fast but arbitrary one to one communication among the processors. Several other families have found their application areas, although they are inferior to ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Comput. Sci., 48:75--94, 1986.

....One class, shuffle type networks, includes the shuffle exchange [116] and de Bruijn [30] networks. A larger class, butterfly type networks, includes the omega, flip, baseline, banyan, and delta networks, as well as several butterfly variants. Descriptions of these networks can be found in [68] or [71] Many of the hypercube networks are computationally equivalent (i.e. they can emulate each other efficiently) Some equivalence results for shuffle type and butterfly type networks can be found in [109] Some of the hypercubic networks are more than just computationally equivalent ....

C. Kruskal and M. Snir, A Unified Theory of Interconnection Network Structure, Theoretical Computer Science, 48(1):75--94, 1986.

....demands that two network topologies be interchangeable for all computational communicational applications. Such equivalence has been established most often using the notion of quasi isometry the inter embeddability of two families of networks with small dilation. One finds in sources such as [2, 8, 14, 17] proofs of the strong equivalence of network topologies such as: paths and rings; meshes and tori; de Bruijn and shuffle exchange networks; butterfly, FFT, and Cube connected cycle networks. Equivalence on important classes of problems. This form of equivalence has received less attention than ....

C.P. Kruskal and M. Snir (1986): A unified theory of interconnection network structure. Theor. Comp. Sci. 48, 75-94.

....which cannot be derived by permuting the bits: 100. Figure 3 is an example of a Delta Network, since all Properties P1 P5 are satisfied. We can also define the path descriptor from a source node to a switch s in stage k, as the string of digits used to get to it from a source node. It is shown in [12] that all paths connecting source nodes to a switch in a Delta Network have the same path descriptor. We thus speak of the path descriptor of s (any switch or destination node) in stage k, and denote it as Z(s) Z 1 (s)Z 2 (s) Z k Gamma1 (s) A Bi Delta Network is a Delta Network in both ....

C.Kruskal & M.Snir, "A Unified Theory of Interconnection Network Structure", Theoretical Computer Science, 48(1):75-94, 1986.

.... multistage network grows as O(N log(N) The best case packaging volume grows as Q(N 3 2 ) and the transit latency grows as Q( p N) like the hypercube [LR86] Quite a variety of networks can be classified as multistage networks including: Butterfly networks, Banyan networks, Bidelta networks [KS86] Benes networks, and Multibutterfly networks. Figures 3.8 through 3.11 show some popular multistage networks. Each stage in these networks routes by successively subdividing the set of possible destinations into a number of equivalence classes equal to the radix of the routing components. For ....

Clyde P. Kruskal and Marc Snir. A Unified Theory of Interconnection Network Structure. In Theoretical Computer Science, pages 75--94, 1986.

....interconnection networks, switching networks, embedding, routing, fault tolerance, multicomputers. y This research was supported by Sprint Corporation. 1 Introduction Many interconnection topologies have been proposed to provide communications on distributed memory parallel processing platforms [3, 8, 12, 20, 21, 25]. Although several of these networks have exhibited acceptable performance, their excessive cost has made them impractical for many parallel applications. As a result, many applications are run on clusters of workstations that communicate through a shared media Local Area Network (LAN) such as ....

Kruskal, C. P. and Snir, M. . "A Unified Theory of Interconnection Network Structure". Theoretical Computer Science, 48:75--94, 1986.

....intermediate stages 1; n Gamma1 allows only 1 connection at any time, where N=2 n is the network size (number of inputs) The Gamma network [9] is an IADM in which all switches in the intermediate stages are 3x3 crossbars. Therefore, it is more powerful than the IADM. Banyan networks [4], such as the Omega( Omega Gamma network [5] and the Generalized Cube Network (GCN) 10] provide a unique path for every input output pair. One can route a permutation by trying to establish a path for each 1 to 1 request in the permutation. This straight forward approach takes O(N log N) time ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....delta networks can be seen as generalizations of the butterfly network, which is known to be equivalent to a shuffle based network of depth lg n (e.g. see [7, Section 3. 8] More precisely, the butterfly network is the unique network that is both a delta network and a reverse delta network [6]. The following recursive definition of a reverse delta network is crucial for understanding our proof technique: A reverse delta network with 2 k 1 inputs and depth k 1 consists of two parallel 2 k input reverse delta networks of depth k, followed by a final level of up to 2 k ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....permutations, Pi (1) Pi (d Gamma1) resulting in a depth log N network with 2d Theta 2d switches. For example, see Figure 4. Butterflies and dilated butterflies are special cases of multibutterflies, and multibutterflies are contained in the much larger class of delta networks [11]. Delta networks are characterized by the up down property of the edges that is used for message routing. The notion of up and down edges can be formalized in terms of splitters. More precisely, the edges from level l to level l 1 in rows jN 2 l to (j 1)N 2 l Gamma 1 in a multibutterfly ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....center is the same, resulting in predictable communication latency between nodes. There exists only one path between any two leaf nodes in a RTN. Because of this property, and the fact that there is a path between every pair of leaf nodes, RTN inherits the desirable properties of Banyan networks[3]. The diameter the longest distance between two leaf nodes of RTN(w; r) is 2r Gamma 1 since the distance from a leaf node to the atomic center is r and the distance from the atomic center to another leaf node is r so there exist 2r Gamma 1 switches in the path. The RTN(w; r) s atomic ....

Clyde P. Kruskal and Marc Snir. A Unified Theory of Interconnection Network Structure. Theoretical Computer Science, 48:75--94, 1986.

....: 41 4.1 Signaling schema for CS Network : 46 4. 2 Path information in Payload of Signaling cell : 49 vi CHAPTER 1 INTRODUCTION Many interconnection topologies have been proposed for building Massively Parallel Processing (MPP) networks [1, 2, 3, 4, 5, 6]. Although many of these networks have been implemented and have performed adequately, their excessive cost has made them impractical for many parallel applications. As a result, many applications are run on clusters of workstations that have traditionally communicated through a shared media Local ....

....center is the same, resulting in predictable communication latency between nodes. There exists only one path between any two leaf nodes in a RT. Because of this property, and the fact that there is a path between every pair of leaf nodes, RT inherits the desirable properties of Banyan networks[2]. The diameter the longest distance between two leaf nodes of RT(w; r) is 2r Gamma 1 since the distance from a leaf node to the atomic center is r and the distance from the atomic center to another leaf node is r so there exist 2r Gamma 1 switches in the path. The RT(w; r) s atomic center ....

C. P. Kruskal and M. Snir, "A Unified Theory of Interconnection Network Structure", Theoretical Computer Science, 48:75--94, 1986.

....n) 2.3 Bidelta Networks There has been a large amount of research on multistage interconnection networks. Kruskal and Snir have found that many of these networks, such as the indirect binary cube (or unfolded butterfly network) the omega network, the SW banyan network, and so on, are isomorphic [8]. That is, one can be produced from the other by simply rearranging the nodes at each stage. These networks are referred to as bidelta interconnection networks. Let n = 2 d ; an n Thetan or d stage bidelta network is composed of d 1 stages of nodes, interconnected by d stages of connections. At ....

C.P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....01 01 01 01 01 11 11 11 11 11 11 11 11 00 00 00 00 00 00 00 00 AA B C Figure 2: An example of RTN ( 4, 3 ) A RTN network has the following properties: 1. It is symmetric. 2. There exists one and only one path between each pair of leaf nodes. This corresponds to the definition of a Bidelta network[9]. 3. The Identical Address Connection Method, developed through this research, can be used to assign addresses to leaf nodes. For instance, leaf node A of RTN(4, 3) in Figure 2 has address 10 01 11. This means the leaf node 10 is attached to port 01 11 of the sub RTN(4,2) which is attached to ....

Clyde P. Kruskal and Marc Snir. A Unified Theory of Interconnection Network Structure. Theoretical Computer Science, 48:75--94, 1986. The paper shows the relationship between the topology of interconnection networks and their functional properties.

....Because it is harder to congest a channel than it is to congest a single wire in a butterfly, dilated butterflies are better routing networks than simple butterflies [8, 9, 16] 1. 2 Delta networks Butterfly and dilated butterfly networks belong to a larger class of networks called delta networks [10]. The switches on each level of a delta network can be partitioned into blocks. All of the switches on level 0 belong to the same block. On level 1, there are two blocks, one consisting of the switches that are in the upper N=2 rows, and the other consisting of the switches that are in the lower ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....example, Figure 3 shows the logical path from any input to output 011. In a butterfly, this logical path specifies a unique path through the network, since only one up and one down edge emanate from each switch. In fact, a splitter network with multiplicity one is very similar to a delta network [17]. In a general splitter network with multiplicity d, however, each switch will have d up and d down edges, and each step of the logical path can be taken on any one of d edges. Hence, one logical path can be realized by a myriad of physical paths in a general splitter network. 1.3 ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

.... factors) for a large variety of leveled hypercube derivative host graphs that we collectively term butterfly like and that include the Benes, cube connected cycles, and multistage cube networks, as well as all bidelta networks (such as the Omega, flip, baseline, and reverse baseline) cf. 4] [18], 23] 27] In fact, versions of our results hold for an even broader class of networks; cf. Section 5. By dint of the efficient embeddings of butterfly like graphs in hypercubes reported in [13] our upper bound results extend verbatim to emulations by hypercubes. In particular: Any maxdegree d ....

C.P. Kruskal and M. Snir (1986): A unified theory of interconnection network structure. Theor. Comp. Sci. 48, 75-94.

....containing, or that would contain, the packet being served is called the head slot, the packet is called the head packet. Earlier analyses and the analysis presented here apply to networks having symmetry properties that are easily proven for a subclass of banyan networks called bidelta networks [10]. A network is a delta network if the module output needed by a packet is uniquely determined only by the stage number and the packet s destination [13] it is a bidelta network if a similar property holds when routing from outputs to inputs. For example, in an n stage omega network using 2 ....

....using 2 Theta 2 modules a packet in stage j uses module output 0 if digit n Gamma j Gamma 1 of its destination is 0 and output 1 otherwise. Most banyans of interest are bidelta networks, and the symmetry properties likely hold for many others. See [14] for an elementary introduction and [10] a discussion of the topology of banyan networks. The network operates in a synchronous clocked fashion; time is divided into cycles. In cycle t a packet can move from a queue in one stage to a queue in the next stage if 1) it is at the head of the queue, 2) it is granted use of the appropriate ....

Kruskal, C.P., and Snir, M. A unified theory of interconnection network structure. Theoretical Computer Science. 48 (1986), 75--94.

....to output 011, which consists of an up edge followed by two down edges. In a butterfly, this logical path specifies a unique path through the network, since only one up and one down edge emanate from each switch. In fact, a splitter network with multiplicity one is very similar to a delta network [5]. In a general splitter network with multiplicity d, however, each switch will have d up and d down edges, and each step of the logical path can be taken on any one of d edges. Hence, one logical path can be realized by a myriad of physical paths in a general splitter network. 2.3 Randomly wired ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....connected by links to module inputs in the next stage. Module outputs in the last stage are connected to network outputs. For a banyan network the links can be connected in any way as long as there is exactly one path between every input output pair [3] See [13] for an elementary introduction and [8] a discussion of the topology of banyan networks. Each module consists of 2 d slot queues, connected to a 2 Theta 2 crossbar switch. Module inputs connect to the queues; queue outputs connect to the crossbar; crossbar outputs connect to the module outputs. A queue s feeder queues are the queues ....

C. P. Kruskal and M. Snir, "A unified theory of interconnection network structure," Theoretical Computer Science, vol. 48, pp. 75--94, 1986.

....as that of the destination node. After that, the next dimension to travel is selected in a fixed order. If links are bidirectional, the direction which leads to the shortest path is selected. 2.1. 2 Multistage and Single Stage Shuffle Exchange Networks A K ary multistage shuffle exchange network [51, 52, 53] consists of dlog K Ne shuffleexchange stages. Each stage has N input and N output terminals that are connected to N output terminals of the previous stage and N input terminals of the next stage, respectively, as shown in Figure 3.1. In a stage, there are N=K crossbar switches of size K, i.e. ....

....the infinite queues shift a part of switch queueing delay to the DMUX queueing delay, this does not affect our conclusions. Chapter 3 Shortest Path Routing in Single Stage Shuffle Exchange Networks 3. 1 Introduction Among various interconnection network architectures, shuffle exchange networks [83, 51, 52, 53] have been one of the most popular architectures. These networks have distributed self routing capabilities, and when network size is increased, the internode distance and the number of switch components are increased logarithmically. These features make shuffle exchange networks suitable for ....

C. P. Kruskal and M. Snir, "A unified theory of interconnection network structure," Theoretical Computer Science, vol. 48, pp. 75--94, Aug. 1986.

....from simulation, message length has a strong effect on performance. Since networks used for communication switches and parallel computers must carry messages having varying lengths, a non uniform message length analysis is needed. Such an analysis had been performed by Kruskal, Snir, and Weiss [4] for infinite buffered networks. The networks they analyzed have output buffered switching elements (SE s) in which queues can be simultaneously fed by any number of SE inputs. Exact first stage switching element queue state distributions were found. An empirically derived formula was then used to ....

C. P. Kruskal and M. Snir, "A unified theory of interconnection network structure," Theoretical Computer Science, vol. 48, pp. 75--94, 1986.

....has N = 2 64 input output terminals 1 Actually, the network, in the form used by Portz and here, is that of Waksman, who invented it in 1964. The network invented by Benes, also in 1964, is called today the Omega network. The topologies seem to be different, but the networks are isomorphic [4]. If f=1 If f=0 Figure 1: The Switching Box. n 1 B n 1 B Figure 2: The recursive structure of the Benes network B n . Figure 3: The B 1 . and 128( 2 Delta 64) columns of switching boxes. The overall number of switching boxes is jSj = 128 Delta 2 63 = 2 70 . Clearly, two difficulties ....

Kruskal, C. P., Snir, M., "A Unified Theory of Interconnection Network Structure", Theoretical Computer Science, Vol. 48, pp. 75--94, 1986.

....shown in Fig. 5(d) Fig. 5(e) is apparently a reverse Omega network (or a flip network [3] Hence the original CCC n is turned into a reverse Omega network with 1 2 n inputs and 1 2 n outputs. Note that the reverse Omega network (with 1 2 n inputs and 1 2 n outputs) has the banyan property [10]: each input node u is connected to each output node v by exactly one path of length s Gamma 1. Let e be a stage k edge of the reverse Omega network where 0 k s Gamma 2. One end point of e reaches precisely 2 s Gammak Gamma2 distinct output nodes while the other endpoint of e reaches ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48(1):75--94, 1986.

....as shown in Fig. 6(d) Fig. 6(e) is a reverse Omega network (or a flip network [3] Hence, the original CCC(n) is transformed into a reverse Omega network with 1 2 n inputs and 1 2 n outputs. Note that the reverse Omega network (with 1 2 n inputs and 1 2 n outputs) has the banyan property [10]: each input node u is connected to each output node v by exactly one path of length s Gamma 1. Let e be a stage k edge of the reverse Omega network, where 1 k s Gamma 1. One end point of e reaches precisely 2 s Gammak Gamma1 distinct output nodes while the other end point of e reaches ....

C.P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....used to make a multibutterfly or multi Benes Figure 7: An 8 input 2 multi Benes network. network can be replaced by any Delta network. A Delta network is a regular network formed by splitters like the butterfly, but for which the individual connections within each splitter can be arbitrary [14]. 3 A proof that the multi Benes network is nonblocking In this section we prove that the multi Benes network is a strict sense nonblocking connector. As a consequence, a simple algorithm like breadth first search can be used to establish a single path from any unused input to any unused output ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

....In particular, each basic butterfly component used to make a multibutterfly or multi Benes network can be replaced by any Delta network. A Delta network is a regular network formed by splitters like the butterfly, but for which the individual connections within each splitter can be arbitrary [14]. Figure 7: An 8 input 2 multi Benes network. 3 A proof that the multi Benes network is nonblocking In this section we prove that the multi Benes network is a strict sense nonblocking connector. As a consequence, a simple algorithm like breadth first search can be used to establish a single ....

C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

No context found.

Kruskal, C. P. and Snir, M. (1986) A unified theory of interconnection network structure. Theoret. Comp. Sci., 48, 75--94.

No context found.

Clyde P. Kruskal and Marc Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

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C. P. Kruskal and M. Snir, A unified theory of interconnection network structure, Theoretical Computer Science, 48 (1986), pp. 75--94.

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C. P. Kruskal and M. Snir. A unified theory of interconnection network structure. Theoretical Computer Science, 48:75--94, 1986.

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C.P. Kruskal and M. Snir, A Unified Theory of Interconnection Network Structure, Ultracomputer note #106, 1986.

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